# Registration beyond Points: General Affine Subspace Alignment via Geodesic Distance on Grassmann Manifold

arXiv: https://arxiv.org/abs/2507.17998 · HTML: https://mlsanity.com/paper/2507.17998/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 47 references that are in the map.
- First posted on arXiv 24 Jul 2025.

## Builds on (most cited first, 15 of 15)

- [ORB-SLAM: a Versatile and Accurate Monocular SLAM System](https://mlsanity.com/paper/1502.00956/) (2015)
- [The Replica Dataset: A Digital Replica of Indoor Spaces](https://mlsanity.com/paper/1906.05797/) (2019)
- [NICE-SLAM: Neural Implicit Scalable Encoding for SLAM](https://mlsanity.com/paper/2112.12130/) (2021)
- [Go-ICP: A Globally Optimal Solution to 3D ICP Point-Set Registration](https://mlsanity.com/paper/1605.03344/) (2016)
- [Learning to Parse Wireframes in Images of Man-Made Environments](https://mlsanity.com/paper/2007.07527/) (2020)
- [GlueStick: Robust Image Matching by Sticking Points and Lines Together](https://mlsanity.com/paper/2304.02008/) (2023)
- [PL-SLAM: a Stereo SLAM System through the Combination of Points and Line Segments](https://mlsanity.com/paper/1705.09479/) (2017)
- [3D Line Mapping Revisited](https://mlsanity.com/paper/2303.17504/) (2023)
- [Scalable Dense Non-rigid Structure-from-Motion: A Grassmannian Perspective](https://mlsanity.com/paper/1803.00233/) (2018)
- [CvxPnPL: A Unified Convex Solution to the Absolute Pose Estimation Problem from Point and Line Correspondences](https://mlsanity.com/paper/1907.10545/) (2019)
- [Global Data Association for SLAM with 3D Grassmannian Manifold Objects](https://mlsanity.com/paper/2205.08556/) (2022)
- [PlueckerNet: Learn to Register 3D Line Reconstructions](https://mlsanity.com/paper/2012.01096/) (2020)
- [Grassmann Manifold Flows for Stable Shape Generation](https://mlsanity.com/paper/2211.02900/) (2022)
- [SLIM: Scalable and Lightweight LiDAR Mapping in Urban Environments](https://mlsanity.com/paper/2409.08681/) (2024)
- [Object Tracking via Non-Euclidean Geometry: A Grassmann Approach](https://mlsanity.com/paper/1403.0309/) (2014)

## Built on by (most cited first, 1 of 1)

- [RoEL: Robust Event-based 3D Line Reconstruction](https://mlsanity.com/paper/2602.18258/) (2026)

## Abstract (arXiv)

Affine Grassmannian has been favored for expressing proximity between lines and planes due to its theoretical exactness in measuring distances among features. Despite this advantage, the existing method can only measure the proximity without yielding the distance as an explicit function of rigid body transformation. Thus, an optimizable distance function on the manifold has remained underdeveloped, stifling its application in registration problems. This paper is the first to explicitly derive an optimizable cost function between two Grassmannian features with respect to rigid body transformation (R and t). Specifically, we present a rigorous mathematical proof demonstrating that the bases of high-dimensional linear subspaces can serve as an explicit representation of the cost. Finally, we propose an optimizable cost function based on the transformed bases that can be applied to the registration problem of any affine subspace. Compared to vector parameter-based approaches, our method is able to find a globally optimal solution by directly minimizing the geodesic distance which is agnostic to representation ambiguity. The resulting cost function and its extension to the inlier-set maximizing Branch-and-Bound (BnB) solver have been demonstrated to improve the convergence of existing solutions or outperform them in various computer vision tasks. The code is available on https://github.com/joomeok/GrassmannRegistration.